Bipolar oriented random planar maps with large faces and exotic SLE${}_{\kappa }(\rho )$ processes

Abstract

We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $\alpha \in (1,2)$ that if the face degree is in the domain of attraction of an $\alpha$-stable Lévy process, the corresponding random planar map has an infinite volume limit in the Benjamini-Schramm topology. We also show in the limit that the properly rescaled contour functions associated with the northwest and southeast trees converge in law to a certain correlated pair of $\alpha$-stable Lévy processes. Combined with other work, this allows us to identify the scaling limit of the planar map with an SLE${}_{\kappa }(\rho )$ process with $\rho =\kappa -4<-2$ on $\sqrt{\kappa }$-Liouville quantum gravity for $\kappa \in (4/3,2)$ where $\alpha ,\kappa$ are related by $\alpha =4/\kappa -1$.

24 pages, 2 figures